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Mirrors > Home > ILE Home > Th. List > ssriv | GIF version |
Description: Inference rule based on subclass definition. (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
ssriv.1 | ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵) |
Ref | Expression |
---|---|
ssriv | ⊢ 𝐴 ⊆ 𝐵 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfss2 2934 | . 2 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)) | |
2 | ssriv.1 | . 2 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵) | |
3 | 1, 2 | mpgbir 1342 | 1 ⊢ 𝐴 ⊆ 𝐵 |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1393 ⊆ wss 2917 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-11 1397 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-in 2924 df-ss 2931 |
This theorem is referenced by: ssid 2964 ssv 2965 difss 3070 ssun1 3106 inss1 3157 unssdif 3172 inssdif 3173 unssin 3176 inssun 3177 difindiss 3191 undif3ss 3198 0ss 3255 difprsnss 3502 snsspw 3535 pwprss 3576 pwtpss 3577 uniin 3600 iuniin 3667 iundif2ss 3722 iunpwss 3743 pwuni 3943 pwunss 4020 omsson 4335 limom 4336 xpsspw 4450 dmin 4543 dmrnssfld 4595 dmcoss 4601 dminss 4738 imainss 4739 dmxpss 4753 rnxpid 4755 enq0enq 6529 nqnq0pi 6536 nqnq0 6539 zssre 8252 zsscn 8253 nnssz 8262 uzssz 8492 divfnzn 8556 zssq 8562 qssre 8565 rpssre 8593 ixxssxr 8769 ixxssixx 8771 iooval2 8784 ioossre 8804 rge0ssre 8846 fzssuz 8928 fzssp1 8930 uzdisj 8955 nn0disj 8995 fzossfz 9021 fzouzsplit 9035 fzossnn 9045 fzo0ssnn0 9071 fclim 9815 bj-omsson 10087 |
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