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| Mirrors > Home > MPE Home > Th. List > psseq1 | Structured version Visualization version GIF version | ||
| Description: Equality theorem for proper subclass. (Contributed by NM, 7-Feb-1996.) |
| Ref | Expression |
|---|---|
| psseq1 | ⊢ (𝐴 = 𝐵 → (𝐴 ⊊ 𝐶 ↔ 𝐵 ⊊ 𝐶)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | sseq1 4009 | . . 3 ⊢ (𝐴 = 𝐵 → (𝐴 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐶)) | |
| 2 | neeq1 3003 | . . 3 ⊢ (𝐴 = 𝐵 → (𝐴 ≠ 𝐶 ↔ 𝐵 ≠ 𝐶)) | |
| 3 | 1, 2 | anbi12d 632 | . 2 ⊢ (𝐴 = 𝐵 → ((𝐴 ⊆ 𝐶 ∧ 𝐴 ≠ 𝐶) ↔ (𝐵 ⊆ 𝐶 ∧ 𝐵 ≠ 𝐶))) |
| 4 | df-pss 3971 | . 2 ⊢ (𝐴 ⊊ 𝐶 ↔ (𝐴 ⊆ 𝐶 ∧ 𝐴 ≠ 𝐶)) | |
| 5 | df-pss 3971 | . 2 ⊢ (𝐵 ⊊ 𝐶 ↔ (𝐵 ⊆ 𝐶 ∧ 𝐵 ≠ 𝐶)) | |
| 6 | 3, 4, 5 | 3bitr4g 314 | 1 ⊢ (𝐴 = 𝐵 → (𝐴 ⊊ 𝐶 ↔ 𝐵 ⊊ 𝐶)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ≠ wne 2940 ⊆ wss 3951 ⊊ wpss 3952 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-cleq 2729 df-ne 2941 df-ss 3968 df-pss 3971 |
| This theorem is referenced by: psseq1i 4092 psseq1d 4095 psstr 4107 sspsstr 4108 brrpssg 7745 sorpssuni 7752 pssnn 9208 marypha1lem 9473 infeq5i 9676 infpss 10256 fin4i 10338 isfin2-2 10359 zornn0g 10545 ttukeylem7 10555 elnp 11027 elnpi 11028 ltprord 11070 pgpfac1lem1 20094 pgpfac1lem5 20099 pgpfac1 20100 pgpfaclem2 20102 pgpfac 20104 islbs3 21157 alexsubALTlem4 24058 wilthlem2 27112 spansncv 31672 cvbr 32301 cvcon3 32303 cvnbtwn 32305 dfon2lem3 35786 dfon2lem4 35787 dfon2lem5 35788 dfon2lem6 35789 dfon2lem7 35790 dfon2lem8 35791 dfon2 35793 lcvbr 39022 lcvnbtwn 39026 mapdcv 41662 |
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