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Mirrors > Home > MPE Home > Th. List > hashrabsn1 | Structured version Visualization version GIF version |
Description: If the size of a restricted class abstraction restricted to a singleton is 1, the condition of the class abstraction must hold for the singleton. (Contributed by Alexander van der Vekens, 3-Sep-2018.) |
Ref | Expression |
---|---|
hashrabsn1 | ⊢ ((#‘{𝑥 ∈ {𝐴} ∣ 𝜑}) = 1 → [𝐴 / 𝑥]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2610 | . 2 ⊢ {𝑥 ∈ {𝐴} ∣ 𝜑} = {𝑥 ∈ {𝐴} ∣ 𝜑} | |
2 | rabrsn 4203 | . 2 ⊢ ({𝑥 ∈ {𝐴} ∣ 𝜑} = {𝑥 ∈ {𝐴} ∣ 𝜑} → ({𝑥 ∈ {𝐴} ∣ 𝜑} = ∅ ∨ {𝑥 ∈ {𝐴} ∣ 𝜑} = {𝐴})) | |
3 | fveq2 6103 | . . . . 5 ⊢ ({𝑥 ∈ {𝐴} ∣ 𝜑} = ∅ → (#‘{𝑥 ∈ {𝐴} ∣ 𝜑}) = (#‘∅)) | |
4 | 3 | eqeq1d 2612 | . . . 4 ⊢ ({𝑥 ∈ {𝐴} ∣ 𝜑} = ∅ → ((#‘{𝑥 ∈ {𝐴} ∣ 𝜑}) = 1 ↔ (#‘∅) = 1)) |
5 | hash0 13019 | . . . . . 6 ⊢ (#‘∅) = 0 | |
6 | 5 | eqeq1i 2615 | . . . . 5 ⊢ ((#‘∅) = 1 ↔ 0 = 1) |
7 | 0ne1 10965 | . . . . . 6 ⊢ 0 ≠ 1 | |
8 | eqneqall 2793 | . . . . . 6 ⊢ (0 = 1 → (0 ≠ 1 → [𝐴 / 𝑥]𝜑)) | |
9 | 7, 8 | mpi 20 | . . . . 5 ⊢ (0 = 1 → [𝐴 / 𝑥]𝜑) |
10 | 6, 9 | sylbi 206 | . . . 4 ⊢ ((#‘∅) = 1 → [𝐴 / 𝑥]𝜑) |
11 | 4, 10 | syl6bi 242 | . . 3 ⊢ ({𝑥 ∈ {𝐴} ∣ 𝜑} = ∅ → ((#‘{𝑥 ∈ {𝐴} ∣ 𝜑}) = 1 → [𝐴 / 𝑥]𝜑)) |
12 | snidg 4153 | . . . . . . . . 9 ⊢ (𝐴 ∈ V → 𝐴 ∈ {𝐴}) | |
13 | 12 | adantr 480 | . . . . . . . 8 ⊢ ((𝐴 ∈ V ∧ {𝑥 ∈ {𝐴} ∣ 𝜑} = {𝐴}) → 𝐴 ∈ {𝐴}) |
14 | eleq2 2677 | . . . . . . . . 9 ⊢ ({𝑥 ∈ {𝐴} ∣ 𝜑} = {𝐴} → (𝐴 ∈ {𝑥 ∈ {𝐴} ∣ 𝜑} ↔ 𝐴 ∈ {𝐴})) | |
15 | 14 | adantl 481 | . . . . . . . 8 ⊢ ((𝐴 ∈ V ∧ {𝑥 ∈ {𝐴} ∣ 𝜑} = {𝐴}) → (𝐴 ∈ {𝑥 ∈ {𝐴} ∣ 𝜑} ↔ 𝐴 ∈ {𝐴})) |
16 | 13, 15 | mpbird 246 | . . . . . . 7 ⊢ ((𝐴 ∈ V ∧ {𝑥 ∈ {𝐴} ∣ 𝜑} = {𝐴}) → 𝐴 ∈ {𝑥 ∈ {𝐴} ∣ 𝜑}) |
17 | nfcv 2751 | . . . . . . . . 9 ⊢ Ⅎ𝑥{𝐴} | |
18 | 17 | elrabsf 3441 | . . . . . . . 8 ⊢ (𝐴 ∈ {𝑥 ∈ {𝐴} ∣ 𝜑} ↔ (𝐴 ∈ {𝐴} ∧ [𝐴 / 𝑥]𝜑)) |
19 | 18 | simprbi 479 | . . . . . . 7 ⊢ (𝐴 ∈ {𝑥 ∈ {𝐴} ∣ 𝜑} → [𝐴 / 𝑥]𝜑) |
20 | 16, 19 | syl 17 | . . . . . 6 ⊢ ((𝐴 ∈ V ∧ {𝑥 ∈ {𝐴} ∣ 𝜑} = {𝐴}) → [𝐴 / 𝑥]𝜑) |
21 | 20 | a1d 25 | . . . . 5 ⊢ ((𝐴 ∈ V ∧ {𝑥 ∈ {𝐴} ∣ 𝜑} = {𝐴}) → ((#‘{𝑥 ∈ {𝐴} ∣ 𝜑}) = 1 → [𝐴 / 𝑥]𝜑)) |
22 | 21 | ex 449 | . . . 4 ⊢ (𝐴 ∈ V → ({𝑥 ∈ {𝐴} ∣ 𝜑} = {𝐴} → ((#‘{𝑥 ∈ {𝐴} ∣ 𝜑}) = 1 → [𝐴 / 𝑥]𝜑))) |
23 | snprc 4197 | . . . . 5 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅) | |
24 | eqeq2 2621 | . . . . . 6 ⊢ ({𝐴} = ∅ → ({𝑥 ∈ {𝐴} ∣ 𝜑} = {𝐴} ↔ {𝑥 ∈ {𝐴} ∣ 𝜑} = ∅)) | |
25 | ax-1ne0 9884 | . . . . . . . . . 10 ⊢ 1 ≠ 0 | |
26 | eqneqall 2793 | . . . . . . . . . 10 ⊢ (1 = 0 → (1 ≠ 0 → [𝐴 / 𝑥]𝜑)) | |
27 | 25, 26 | mpi 20 | . . . . . . . . 9 ⊢ (1 = 0 → [𝐴 / 𝑥]𝜑) |
28 | 27 | eqcoms 2618 | . . . . . . . 8 ⊢ (0 = 1 → [𝐴 / 𝑥]𝜑) |
29 | 6, 28 | sylbi 206 | . . . . . . 7 ⊢ ((#‘∅) = 1 → [𝐴 / 𝑥]𝜑) |
30 | 4, 29 | syl6bi 242 | . . . . . 6 ⊢ ({𝑥 ∈ {𝐴} ∣ 𝜑} = ∅ → ((#‘{𝑥 ∈ {𝐴} ∣ 𝜑}) = 1 → [𝐴 / 𝑥]𝜑)) |
31 | 24, 30 | syl6bi 242 | . . . . 5 ⊢ ({𝐴} = ∅ → ({𝑥 ∈ {𝐴} ∣ 𝜑} = {𝐴} → ((#‘{𝑥 ∈ {𝐴} ∣ 𝜑}) = 1 → [𝐴 / 𝑥]𝜑))) |
32 | 23, 31 | sylbi 206 | . . . 4 ⊢ (¬ 𝐴 ∈ V → ({𝑥 ∈ {𝐴} ∣ 𝜑} = {𝐴} → ((#‘{𝑥 ∈ {𝐴} ∣ 𝜑}) = 1 → [𝐴 / 𝑥]𝜑))) |
33 | 22, 32 | pm2.61i 175 | . . 3 ⊢ ({𝑥 ∈ {𝐴} ∣ 𝜑} = {𝐴} → ((#‘{𝑥 ∈ {𝐴} ∣ 𝜑}) = 1 → [𝐴 / 𝑥]𝜑)) |
34 | 11, 33 | jaoi 393 | . 2 ⊢ (({𝑥 ∈ {𝐴} ∣ 𝜑} = ∅ ∨ {𝑥 ∈ {𝐴} ∣ 𝜑} = {𝐴}) → ((#‘{𝑥 ∈ {𝐴} ∣ 𝜑}) = 1 → [𝐴 / 𝑥]𝜑)) |
35 | 1, 2, 34 | mp2b 10 | 1 ⊢ ((#‘{𝑥 ∈ {𝐴} ∣ 𝜑}) = 1 → [𝐴 / 𝑥]𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∨ wo 382 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 {crab 2900 Vcvv 3173 [wsbc 3402 ∅c0 3874 {csn 4125 ‘cfv 5804 0cc0 9815 1c1 9816 #chash 12979 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-hash 12980 |
This theorem is referenced by: rusgrasn 26472 |
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