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Mirrors > Home > MPE Home > Th. List > numthcor | Structured version Visualization version GIF version |
Description: Any set is strictly dominated by some ordinal. (Contributed by NM, 22-Oct-2003.) |
Ref | Expression |
---|---|
numthcor | ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 ∈ On 𝐴 ≺ 𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq1 4586 | . . 3 ⊢ (𝑦 = 𝐴 → (𝑦 ≺ 𝑥 ↔ 𝐴 ≺ 𝑥)) | |
2 | 1 | rexbidv 3034 | . 2 ⊢ (𝑦 = 𝐴 → (∃𝑥 ∈ On 𝑦 ≺ 𝑥 ↔ ∃𝑥 ∈ On 𝐴 ≺ 𝑥)) |
3 | vpwex 4775 | . . . 4 ⊢ 𝒫 𝑦 ∈ V | |
4 | 3 | numth2 9176 | . . 3 ⊢ ∃𝑥 ∈ On 𝑥 ≈ 𝒫 𝑦 |
5 | vex 3176 | . . . . . 6 ⊢ 𝑦 ∈ V | |
6 | 5 | canth2 7998 | . . . . 5 ⊢ 𝑦 ≺ 𝒫 𝑦 |
7 | ensym 7891 | . . . . 5 ⊢ (𝑥 ≈ 𝒫 𝑦 → 𝒫 𝑦 ≈ 𝑥) | |
8 | sdomentr 7979 | . . . . 5 ⊢ ((𝑦 ≺ 𝒫 𝑦 ∧ 𝒫 𝑦 ≈ 𝑥) → 𝑦 ≺ 𝑥) | |
9 | 6, 7, 8 | sylancr 694 | . . . 4 ⊢ (𝑥 ≈ 𝒫 𝑦 → 𝑦 ≺ 𝑥) |
10 | 9 | reximi 2994 | . . 3 ⊢ (∃𝑥 ∈ On 𝑥 ≈ 𝒫 𝑦 → ∃𝑥 ∈ On 𝑦 ≺ 𝑥) |
11 | 4, 10 | ax-mp 5 | . 2 ⊢ ∃𝑥 ∈ On 𝑦 ≺ 𝑥 |
12 | 2, 11 | vtoclg 3239 | 1 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 ∈ On 𝐴 ≺ 𝑥) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 ∃wrex 2897 𝒫 cpw 4108 class class class wbr 4583 Oncon0 5640 ≈ cen 7838 ≺ csdm 7840 |
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-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-ac2 9168 |
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-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 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-se 4998 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-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-isom 5813 df-riota 6511 df-wrecs 7294 df-recs 7355 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-card 8648 df-ac 8822 |
This theorem is referenced by: cardmin 9265 |
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