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Mirrors > Home > ILE Home > Th. List > sseqtrrd | GIF version |
Description: Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.) |
Ref | Expression |
---|---|
sseqtrrd.1 | ⊢ (𝜑 → 𝐴 ⊆ 𝐵) |
sseqtrrd.2 | ⊢ (𝜑 → 𝐶 = 𝐵) |
Ref | Expression |
---|---|
sseqtrrd | ⊢ (𝜑 → 𝐴 ⊆ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sseqtrrd.1 | . 2 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) | |
2 | sseqtrrd.2 | . . 3 ⊢ (𝜑 → 𝐶 = 𝐵) | |
3 | 2 | eqcomd 2145 | . 2 ⊢ (𝜑 → 𝐵 = 𝐶) |
4 | 1, 3 | sseqtrd 3135 | 1 ⊢ (𝜑 → 𝐴 ⊆ 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 ⊆ wss 3071 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-11 1484 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-in 3077 df-ss 3084 |
This theorem is referenced by: sseqtrrid 3148 fnfvima 5652 tfrlemiubacc 6227 tfr1onlemubacc 6243 tfrcllemubacc 6256 rdgivallem 6278 nnnninf 7023 dfphi2 11896 ctinf 11943 toponss 12193 ssntr 12291 iscnp3 12372 cnprcl2k 12375 tgcn 12377 tgcnp 12378 ssidcn 12379 cncnp 12399 txcnp 12440 imasnopn 12468 hmeontr 12482 blssec 12607 blssopn 12654 xmettx 12679 metcnp 12681 nnsf 13199 nninfsellemsuc 13208 |
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